Entropy conserving implicit time integration in a Discontinuous Galerkin solver in entropy variables

This article presents a fully discrete entropy conserving/stable method based on a Discontinuous Galerkin (DG) discretization in entropy variables coupled with a modified Crank-Nicolson scheme. The entropy conserving time integration is inspired by the work of LeFloch, originally developed in the context of a Finite Volume method in conservative variables. This entropy conserving time integrator is here adapted to a DG discretization in entropy variables also demonstrating the fulfilment of entropy conservation regardless of the time step size and the type of elements used (quadrangular or triangular elements, possibly with curved edges). The performance of the implicit method will be demonstrated by computing several inviscid flow problems, i.e., the convection of an isentropic vortex, the double shear layer, the Kelvin-Helmholtz instability, the shedding flow past a triangular wedge, the Sod shock tube, the receding flow and the Taylor-Green vortex.